Empirical Modls for Dark Matter Halos

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EMPIRICAL MODELS FOR DARK MATTER HALOS. I. NONPARAMETRIC CONSTRUCTIONOF DENSITY PROFILES AND COMPARISON WITH PARAMETRIC MODELS

David MerrittDepartment of Physics, Rochester Institute of Technology, Rochester, NY 14623

Alister W. Graham

1

Mount Stromlo and Siding Spring Observatories, Australian National University, Weston Creek, ACT 2611, Australia

Ben MooreUniversity of Zurich, CH-8057 Zurich, Switzerland

Jurg DiemandDepartment of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064

and

Balsa TerzicDepartment of Physics, Northern Illinois University, DeKalb, IL 60115

Received 2006 May 11; accepted 2006 September 1

ABSTRACT

We use techniques from nonparametric function estimation theory to extract the density profiles, and their deriva-tives, from a set ofN-body dark matter halos. We consider halos generated from CDM simulations of gravitationalclustering, as well as isolated spherical collapses. The logarithmic density slopes dlog /dlog r of the CDMhalos are found to vary as powerlaws in radius, reachingvalues of% 1 at the innermost resolved radii, $102rvir.This behavior is significantly different from that of broken-power-law models like the Navarro-Frenk-White ( NFW)profile but similar to that of models like de Vaucouleurss. Accordingly, we compare the N-body density profiles withvarious parametric models to find which provide the best fit. We consider an NFW-like model with arbitrary innerslope; Dehnen & McLaughlins anisotropic model; Einastos model (identical in functional form to Sersics modelbut fitted to the space density); and the density model of Prugniel & Simien that was designed to match the depro-jected form of Sersics R1

=n law. Overall, the best-fitting model to theCDM halos is Einastos, although the Prugniel-Simien and Dehnen-McLaughlin models also perform well. With regard to the spherical-collapse halos, both thePrugniel-Simien and Einasto models describe the density profiles well, with an rms scatter some 4 times smaller thanthat obtained with either the NFW-like model or the three-parameter Dehnen-McLaughlin model. Finally, we confirmrecent claims of a systematic variation in profile shape with halo mass.

Key words: dark matter galaxies: halos methods: n-body simulations

1. INTRODUCTION

A fundamental question is the distribution of matter in boundsystems (galaxies, galaxy clusters, and dark matter halos) thatformin an expanding universe. Early work on the self-similar collapseof (spherical) primordial overdensities resulted in virialized struc-tures having density profiles described by a single power law(e.g., Fillmore & Goldreich 1984; Bertschinger 1985; Hoffman1988). Some of the firstN-body simulations were simple cold-collapse calculations like these (e.g., van Albada 1961; Aarseth1963; Henon 1964; Peebles 1970). It was quickly realized thatgiven appropriately low but nonzero levels of initial random ve-

locity, the end state of such systems departed from a simple powerlaw, resembling instead the de Vaucouleurs (1948) R1=4 profilesobserved in elliptical galaxies (e.g., van Albada 1982; Aguilar& Merritt 1990). A closer reinspection of the data, however(e.g., Figs. 46 in van Albada 1982; Fig. 4 in Carlberg et al.1986) reveals obvious and systematic deviations from the R1/4

model in the cold collapses (see also Nipoti et al. 2006). Froma visual inspection of these figures, one can see that the dis-tributions would be better described with an R1/n profile withn < 4.

AsN-body techniques improved, the logarithmic profile slopesof cold dark matter (CDM) halos, simulated in hierarchical mergermodels, were also observed to steepen with increasing radius(e.g., West et al. 1987; Frenk et al. 1988; Efstathiou et al. 1988).Dubinski & Carlberg (1991) adopted Hernquists (1990) double-power-law model (itself a modification of Jaffes [1983] model)to describe these density profiles. This empirical model has aninner logarithmic slope of1 and an outer logarithmic slope of4. It was introduced as an analytical approximation to the de-projected form of de Vaucouleurs (1948) profile. Navarro et al.(1995) modified this to give the so-called Navarro-Frenk-White

(NFW) model, which has an outer logarithmic slope of3 ra-ther than 4, while Mooreet al. (1998, 1999) suggested that a fur-ther variation having an inner logarithmic slope of1.4 or1.5might be more appropriate.

The density profiles of N-body halos typically span only $2decades in radius, between the virial radius and an inner limit setby the N-body resolution. It has long been clear that other func-tional forms might fit such limited data as well or better than theNFW or Moore profiles. Recently, Navarro et al. (2004) arguedfor a model, like de Vaucouleurss in which the logarithmic slopevaries continuously with radius:

dln

dln r 2

r

r2

; 11 Current address: Centre for Astrophysics and Supercomputing, Swinburne

University of Technology, Hawthorn, VIC 3122, Australia.

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# 2006. The American Astronomical Society. All rights reserved. Printed in U.S.A.


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i.e.,

(r) / exp Ar ; 2

where r2 is the radius at which the logarithmic slope of the den-sity is 2 and is a parameter describing the degree of curvatureof the profile. Merritt et al. (2005) pointed out that this is thesame relation between slope and radius that defines Sersics (1963,

1968) model, with the difference that Sersics model is tradition-ally applied to the projected (surface) densities of galaxies, not tothe space density. Merritt et al. further showed that the (space)density profiles of a sample ofN-body halos were equally well fit-tedby equation (2) or by a deprojected Sersic profile, and that bothof these models provide better fitsthanan NFW-like, double-power-law model with a variable inner slope. Hence, Sersics modelthe function that is so successful at describing the luminosity pro-files of early-type galaxies and bulges (e.g., Caon et al. 1993;Graham & Guzman 2003; and references therein) and the projecteddensity of hot gas in galaxy clusters (Demarco et al. 2003) is alsoan excellent description ofN-body halos.

(To limit confusion, we henceforth refer to eq. [2] as Einastosr1

=n model when applied to space density profiles and as Sersics

R1=n

model when applied to projected density profiles, with Rthe radius on the plane of the sky. The former name acknowl-edges Einastos [1965, 1968, 1969] early and extensive use ofeq. [2] to model the light and mass distributions of galaxies [seealso Einasto & Haud 1989]. In addition, we henceforth replacethe exponent by 1/n in keeping with the usage established bySersic and de Vaucouleurs.)

In this paper we continue the analysis of alternativesto the NFWand Moore profiles, using a new set ofN-body halos. Among thevarious models that we consider is the Prugniel-Simien (Prugniel& Simien 1997) law, first developed as an analytic approximationto the deprojected form of the Sersic R1

=n profile. Apart from thework of Lima Neto et al. (1999), Pignatelli & Galletta (1999),and Marquez et al. (2000, 2001), the Prugniel-Simien model has

received little attention to date. Demarco et al. (2003) have, how-ever, applied it to the gas density profiles of 24 galaxy clustersobserved withROSAT, and Terzic & Graham (2005) showed thatit provides a superior description of the density profiles of realelliptical galaxies compared with either the Jaffe or Hernquist mod-els. As far as we are aware, ours is the first application of thePrugniel-Simien model to N-body halos.

As in Merritt et al. (2005), we base our model evaluations onnonparametricrepresentations of theN-body density profiles. Suchrepresentations are optimum in terms of their bias-variancetradeoff but are also notable for their flexibility. Not only do they(1) constitute stand-alone, smooth, and continuous represen-tations of the density and its slope, they are also well suited to(2) inferring best-fit values for the fitting parameters of parametric

functions and (3) comparing the goodness of fit of different pa-rametric models via the relative values of the integrated squareerror or a similar statistic. The more standard technique of com-puting binned densities is suitable (although inferior) for (2) and(3) but not for (1), since the density is given only at a discrete setof points and the derivatives are poorly defined, while techniqueslike Sarazins (1980) maximum likelihood algorithm provide a(perhaps) more direct route to (2) but are not appropriate for(1) or (3). Recently, nonparametric function estimation methodshave been applied to many other problems in astrophysics, in-cluding reconstruction of the cosmic microwave background fluc-tuation spectrum ( Miller et al. 2002), dynamics of dwarf galaxies(Wang et al. 2005), and reconstruction of dark matter distributionsvia gravitational lensing (Abdelsalam et al. 1998). Application of

nonparametric methods to the halo density profile problem is per-haps overdue, especially given the importance of determining theinner density slope (Diemand et al. 2005).

In x 2 we introduce the data sets to be analyzed. These consistofN-body simulations of 10 CDM halos and two halos formedby monolithic (nearly spherical) collapse. (Moore et al. [1999]have discussed the similarity between the end state of cold-collapsesimulations and hierarchical CDM models.) In x 3 we present the

nonparametric method used to construct the density profiles andtheir logarithmic slopes. Section 4 presents four different three-parameter models, and x 5 reports how well these empirical mod-els perform. Our findings are summarized in x 7.

In Paper II of this series (Graham et al. 2006a) we explore theEinasto and Prugniel-Simien models in more detail. Specifically,we explore the logarithmic slope of these models and comparethe results with observations of real galaxies. We also present themodels circular velocity profiles and their/3profiles. Helpfulexpressions for the concentration and assorted scale radii (rs, r2,re,Re, rvir, and rmax, the radius where the circular velocity profilehas its maximum value) are also derived. Because the Prugniel-Simien model yields the same parameters as those coming fromSersic-model fits, we are able to show in Paper III (Graham et al.

2006b) the location of our dark matter halos on the Kormendydiagram (e vs. log Re), along with real galaxies. In addition, weshow in Paper III the location of our dark matter halos and realgalaxies and clusters in a new log (e) log (Re) diagram.

2. DATA: DARK MATTER HALOS

We use a sample of relaxed, dark matter halos from Diemandet al. (2004a, 2004b). Details about the simulations, convergencetests, and an estimate of the converged scales can be found inthose papers. Briefly, the sample consists of six cluster-sizedhalos (models A09, B09, C09, D12, E09, and F09) resolved with525 million particles within the virial radius, and four galaxy-sized halos (models G00, G01, G02, and G03) resolved with 24 million particles. The innermost resolved radii are 0.3%0.8%

of the virial radius, rvir. The outermost data point is roughly at thevirial radius, which is defined in such a way that the mean den-sity within rvir is 178

0:45M crit 98:4crit (Eke et al. 1996) using

m 0:268 (Spergel et al. 2003). The virial radius thus enclosesan overdensity that is 368 times denser than the mean matter den-sity. We adopted the same estimates of the halo centers as in theDiemand et al. papers; these were computed using SKID (Stadel2001), a kernel-based routine.

In an effort to study the similarities between cold, collisionlesscollapse halos and CDM halos, we performed two additionalsimulations. We distributed 107 particles with an initial densityprofile (r) / r1, within a unit radius sphere with total mass1 (M11) and 0.1 (M35). The particles were given zero kineticenergy, and the gravitational softening was set to 0.001. Eachsystem collapsed and experienced a radial-orbit instability( Merritt& Aguilar 1985) that resulted in a virialized, triaxial/prolate struc-ture. The lower mass halo, M35, collapsed less violently over alonger period of time.

3. NONPARAMETRIC ESTIMATION OF DENSITYPROFILES AND THEIR DERIVATIVES

Density profiles of N-body halos are commonly constructedby counting particles in bins. While a binned histogram is a bonafide nonparametric estimate of the true density profile, it hasmany undesirable properties; e.g., it is discontinuous, and it de-pends sensitively on the chosen size and location of the bins (see,e.g., Stepanas & Saha 1995). A better approach is to view the

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particle positions as a random sample drawn fromsome unknown,smooth density (r), and to use techniques from nonparametricfunction estimation to construct an estimate of (e.g., Scott1992). In the limit that the sample sizeNtends to infinity, suchan estimate exactly reproduces the density function from whichthe data were drawn, as well as many properties of that function,e.g., its derivatives (Silverman 1986).

We used a kernel-based algorithm for estimating (r), similar

to the algorithms described in Merritt & Tremblay (1994) andMerritt (1996). The starting point is an estimate of the three-dimensional density obtained by replacing each particle at posi-tion ri by a kernel of width hi, and summing the kernel densities:

r XNi1

mi

h3iK

1

hir rij j

: 3

Here mi is the mass associated with the ith particle andKisa nor-malized kernel function, i.e., a density function with unit volume.We adopted the Gaussian kernel,

K(y) 1

(2)3=2ey

2=2: 4

The density estimate of equation (3) has no imposed symme-tries. We now suppose that(r) (r), i.e., that the underlyingdensity is spherically symmetric about the origin. In order for thedensity estimate to have this property, we assume that each par-ticle is smeared uniformly around the surface of the sphere whoseradius is ri. The spherically symmetrized density estimate is

(r) XNi1

mi

h3i

1

4

Zd

Zd sin K

d

hi

; 5a

d

2

r

r

ij j

2

5b r2i r

2 2rri cos ; 5c

where is defined (arbitrarily) from the ri-axis. This can be ex-pressed in terms of the angle-averaged kernel K,

K r; ri; hi 1

4

Z=2=2

d

;

Z20

d sin K h1i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2i r

2 2rri cos q

6a

1

2Z1

1 dK h1i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir

2i r

2

2rriq

; 6b

as

(r) XNi1

mi

h3i

K r; ri; hi : 7

Substituting for the Gaussian kernel, we find

K r; ri; hi 1

(2)3=2rri

h2i

1exp

r2i r2

2h2i

sinh

rri

h2i

:

8

A computationally preferable form is

K 1

2(2)3=2rri

h2i

1exp

(ri r)2

2h2i

! exp

(ri r)2

2h2i

!& ':

9

Equations (7) and (9) define the density estimate. Typically, one

sets up a grid in radius and evaluates (r) discretely on the grid.However, we stress that the density estimate itself is a continuousfunction and is defined independently of any grid.

Given a sample of Npositions and particle masses drawn ran-domly from some (unknown) (r), the goal is to construct an es-timate (r) that is as close as possible, in some sense, to (r).In the scheme just described, one has the freedom to adjust the Nkernel widths hi in order to achieve this. In general, if the hi aretoo small, the density estimate will be noisy, i.e., (r) will ex-hibit a large variance with respect to the true density, while if thehi are too large, the density estimate will be oversmoothed, i.e.,there will be a large bias. (Of course the same is true for binnedhistograms, although in general the bias-variance tradeoff for his-tograms is less good than for kernel estimates.) If the true (r)

were known a priori, one could adjust the hi so as to minimize(for instance) the mean square deviation between (r) and (r).Since (r) is not known a priori for our halos, some algorithmmust be adopted for choosing the hi . We followed the standard practice (e.g., Silverman 1986, p. 101)of varying thehi as a powerof the local density:

hi h0 pilot ri =g

; 10

where pilot(r) is a pilot estimate of(r), and gis the geometricmean of the pilot densities. Since the pilot estimate is used onlyfor assigning the hi, it need not be differentiable, and we con-structed it using a nearest-neighbor scheme.

The final density estimate (r) is then a function of two quan-tities: h0 and . Figure 1 illustrates the dependence of(r) onh0 when the kernel algorithm is applied to a random sample of106 equal-mass particles generated from an Einasto density pro-file with n 5, corresponding to typicalvalues observed in Merrittet al. (2005). Each of the density profile estimates of Figure 1 used 0:3. As expected, for small h0 the estimate of(r) is noisybut faithful in an average way to the true profile, while for largeh0 (r) is a smooth function but is biased at small radii due to theaveraging effect of the kernel. For 0:3 the optimum h0 forthis sample is $0.05re , where re is the half-mass radius comingfrom the Einasto model (see x 4.2).

In what follows we compare the nonparametric estimates (r)derived from the N-body models with various parametric fitting

functions in order to find the best-fitting parameters of the latterby minimizing the rms residuals between the two profiles. Forthis purpose, any of the density estimates in Figure 1 would yieldsimilar results, excepting perhaps the density estimate in the toppanel, which is clearly biased at small radii. In addition, we alsowish to characterize the rms value of the deviation between thetrue profile and the best-fitting parametric models. Here it isuseful for the kernel widths to be chosen such that the residualsare dominated by the systematic differences between the paramet-ric and nonparametric profiles, and not by noise in (r) resultingfrom overly small kernels. We verified that this condition waseasily satisfied for all of the N-body models analyzed here: therewas always found to be a wide range of (h0, ) values such thatthe residuals between (r) and the parametric function were nearly

EMPIRICAL MODELS FOR DARK MATTER HALOS. I. 2687 No. 6, 2006


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constant. This is a consequence of the large particle numbers

(>106) in the N-body models, which imply a low variance evenfor small h0.

As discussed above, quantities like the derivative of thedensity can also be computed directly from (r). Figure 2 showsnonparametric estimates of the slope, dlog /dlog r, for thesame 106 particle data set as in Figure 1. We computed deriva-tives simply by numerically differentiating (r); alternatively,we could have differentiated equation (9). Figure 2 shows that ash0 is increased, the variance in the estimated slope drops, and forh0 % 0:2re the estimate is very close to the true function. We notethat the optimal choice ofh0 when estimating derivatives is largerthan when estimating (r) ($0.2re vs. $0.05re); this is a well-known consequence of the increase in noise associated withdifferentiation. Figure 2 also illustrates the important point that

there is no need to impose an additional level of smoothing whencomputing the density derivatives (as was done, e.g., in Reed et al.2005); it is sufficient to increase h0.

3.1. Application to the N-Body Halos

Figure 3 shows, using 0:3 and h0 0:05re (left) and 0:4 and h0 0:05re (right),

2 the nonparametric estimates of(r)(left) and (r) dlog /dlog r (right) for the 10 N-body halos.Figure 4 shows the same quantities for the two data sets gener-

ated from cold collapses. We stress that these plotsespecially

the derivative plotscould not have been made from tables ofbinned particle numbers. For most profiles the slope is a rathercontinuous function of radius and does not appear to reach anyobvious,asymptotic, central value by $0.01rvir. Instead, (r) variesapproximately as a power ofr; i.e., log versus log ris approx-imately a straight line. Accordingly, we have fitted straight lines,via a least-squares minimization, to the logarithmic profile slopesin the right panels of Figures 3 and 4. The regression coefficients,i.e., slopes, are inset in each panel. (These slope estimates shouldbe seen as indicative only; they are superseded by the model fitsdiscussed below.) In passing we note that such a power-law de-pendence ofon ris characteristic of the Einasto model, with thelogarithmic slope equal to the exponent 1/n. Noise and probable(small) deviations from a perfect Einasto r1

=n model are expected

to produce slightly different exponents when we fit the densityprofiles in the following section with Einastos r1

=n model and anumber of other empirical functions.

The slope at the innermost resolved radius is always close to1, which is also the slope atr 0 in the NFW model. How-ever, there is no indication in Figure 3 that (r) is flattening atsmall radii; i.e., it is natural to conclude that N-body simulationsof higher resolution would exhibit smaller inner slopes. On aver-age, the slope atrvir is around 3, but there are large fluctuations,andsome halos reacha value of4, as previously notedin Diemandet al. (2004b). The reason for these fluctuations may be becausethe outer parts are dynamically very young (i.e., measured in localdynamical times), and they have only partially completed their

2 We have intentionally used a small value of h0 to avoid any possibility ofbiasing the slope estimates.

Fig. 1.Nonparametric bias-variance tradeoff in the estimation of(r) usinga single sample of 106 radii generatedfrom a halo havingan Einasto r1=n density

profile with n 5 (see x 4.2). From top to bottom, h0 (0:1; 0:03; 0:01; 0:003;0:001)re; all estimates use 0:3 (see eqs. [7], [9], and [10]).

Fig. 2.Five estimates of the logarithmic slope of an Einasto r1=n halo,

derived via differentiation of (r). The same sample of 106 radii was used as inFig. 1. From top to bottom, h0 (0:3; 0:2; 0:1; 0:05; 0:03)re; all estimates use 0:4 (see eq. [10]). Dashed lines show the true slope.



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evolution to an equilibrium configuration. We are not able to saywith any confidence what the slopes do beyond rvir.

4. EMPIRICAL MODELS

In this section we present four parametric density models, eachhaving three independent parameters: two scaling parameters

and one shape parameter. We measured the quality of each pa-rametric models fit to the nonparametric (r) values using a stan-dard metric, the integrated square deviation,Z

d log r log(r) log param(r)

2; 11

Fig. 3.Nonparametric estimates of the density (r) (left) and the slope dlog /dlog r(right) for the 10N-body halos of Table 1. Thevirialradius rviris markedwithan arrow. Dashed lines in the right panels are linear fits of log (dlog /dlog r) to log r; regression coefficients are also given.



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where param is understood to depend on the various fitting pa-rameters, as well as on r. Equation (11) is identical in form to theCramervon Mises statistic (e.g., Cox & Hinkley 1974, eq. [6]),an alternative to the Kolmogorov-Smirnov statistic for compar-

ing two (cumulative) distribution functions.We chose to evaluate this integral by discrete summation ona grid spaced uniformly in log r; our measure of goodness of fit(which was also the quantity that was minimized in determiningthe best-fit parameters) was

2

Pmj1

2j

m 3; 12a

j log rj

param rj

" #; 12b

with m 300. With such a large value ofm the results obtainedby minimizing equations (12a) and (11) are indistinguishable.

We note that the quantity 2 in equation (12a) is reminiscent ofthe standard 2, but the resemblance is superficial. For instance,

2 as defined here is independent ofm in the large-m limit (andour choice ofm 300 puts us effectively in this limit). Further-more, there is no binning involved in the computation of2; thegrid is simply a numerical device used in the computation ofequation (11).

4.1. Double-Power-Law Models

Hernquist (1990, his eq. [43]) presented a five-parameter gen-eralization of Jaffes (1983) double-power-law model. Sometimesreferred to as the (, , ) model, it can be written as

(r) s2()= r

rs

1 rrs

!(

)=; 13

where s is the density at the scale radius rs, which marks thecenter of the transition region between the inner and outer powerlaws having slopes ofand , respectively. The parametercontrols the sharpness of the transition (see Zhao1996; Kravtsovet al. 1998; and eqs. [37] and [40b] in Dehnen & McLaughlin2005). Setting (;;) (1; 3; 1) yields the NFW model, while(1.5, 3, 1.5) gives the model in Moore et al. (1999). Other com-binations have been used: for example, (1, 3, 1.5) was appliedin Jing & Suto (2000), and (1, 2.5, 1) was used by Rasia et al.(2004).

In fitting dark matter halos, Klypin et al. (2001, their Fig. 8)have noted a certain degree of degeneracy when all five param-eters are allowed to vary. Graham et al. (2003, theirFigs. 3 and 4)have also observed the parameters of this empirical model to be

highly unstable when applied to (light) profiles having a contin-uously changing logarithmic slope. Under such circumstances,the parameters can be a strong function of the fitted radial extentrather than reflecting the intrinsic physical properties of the pro-file under study. This was found to be the case for the dark matterhalos under study here. We have therefore chosen to constraintwo of the model parameters, holding fixed at 1 and fixedat 3.

In recent years, as the resolution in N-body simulations hasimproved, Moore and collaborators have found that the inner-most (resolved) logarithmic slope of dark matter halos has arange of values that are typically shallower than 1.5, recentlyobtaining a mean value (plus or minus a standard deviation) equalto 1:26 0:17 at 1% of the virial radius (Diemand et al. 2004b).

At the same time, Navarro et al. (2004) report that the NFWmodel underestimates the density over the inner regions of mostof their halos, which have innermost resolved slopes ranging from1.6 to 0.95 (their Fig. 3). A model with an outer slope of3and an inner slope of might therefore be more appropriate.Such a model has been used before and can be written as

(r) 23s

(r=rs)(1 r=rs)

3: 14

The total mass of this model is infinite, however.We have applied the above (1, 3, ) model to our dark matter

density profiles, the results of which are shown in Figure 5 for

the N-body halos, and in the top panel of Figure 6 for the cold-collapse models. The rms scatter is inset ineach figure and ad-ditionally reported in Table 1.

4.1.1. Two-Parameter Models

Recognizing that galaxies appear to have flat inner density profiles (e.g., Flores & Primack 1994; Moore 1994), Burkert(1995)cleverly introduced a density model having an inner slopeof zero and an outer profile that decayed as r3. His model isgiven by the expression

(r) 0r

3s

(r rs)(r2 r2s ); 15

Fig. 4.Nonparametric estimates of (r) (left) and dlog /dlog r (right) for the two collapse models. Dashed lines in the right panels are linear fits oflog (dlog /dlog r) to log r.



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where 0 is the central density and rs is a scale radius. Appli-cation of this model in Figure 7 reveals that, with only two freeparameters, it does not provide as good a fit to the simulateddarkmatter halos as the (1, 3, ) model presented above. The hump-shaped residual profiles in Figure 7 signify the models inabilityto match the curvature of our density profiles. (It is important topoint out that Burkerts model was introduced to fit the observedrotation curves in low surface brightness galaxies after the contri-bution from the baryonic component had been subtracted out, atask that it performs well.) This and other two-parameter modelsare shown in Table 2.

As noted previously, the NFW (;;) (1; 3; 1) model also

has only two parameters: s and rs. Because this model is stilloften used, we apply it to our halos in Figure 8. Comparison withFigure 5 reveals that the NFW model never performs as well asthe (1, 3,) model; the residuals are $50% larger and sometimestwice as large. Importantly, the large-scale curvature observedin many of the NFW residual profiles (Fig. 8) reveals that thismodel does not describe the majority of the halos, and that the(1, 3, ) model should be preferred over the NFW model.

An alternative two-parameter expression has recently beenstud-ied by Dehnen & McLaughlin (2005, their eq. [20b]; see alsoAustin et al. 2005). It is a special case of a more general familyof models, which we test next, when the velocity ellipsoid at thehalo center is isotropic and /3r is a (special) power law in ra-dius, varying as r35=18. This two-parameter density model is an

(;;) (4/9; 31/9; 7/9) model given by

(r) 26s

(r=rs)7=9 1 r=rs

4=9h i6 ; 16

and is applied in Figure 9. It clearly provides a much better matchto the dark matter halo density profiles in comparison with theprevious two-parameter model over the fitted radial range, butthe rms scatter reveals that it does not perform as well as the (1,3, ) model, nor can it describe the spherical collapse halos(Fig. 6). Therefore, in x 4.1.2 we test the more general three-parameter model given in Dehnen & McLaughlin (2005).

4.1.2. Dehnen-McLaughlin Anisotropic Three-Parameter Model

Dehnen & McLaughlin (2005, their eq. [46b]) present a the-oretically motivated, three-parameter model such that

;; 2(2 0)=9; (31 20)=9; (7 100)=9;

and the term 0 reflects the central (r 0) anisotropy: a measureof the tangential to radial velocity dispersion.3 Setting 0 (7100)/9, we have ;; (3 0)/5; (18 0)/5; 0, andtheir density model can be written as

(r) 26s

r=rs 0

1 r=rs 30 =5

h i6 ; 17

As shown in Figure 10, for three of the six cluster-sized halosthis model has the greatest residual scatter of the four differentthree-parameter models tested here. For another two of the sixcluster-sized halos it has the second greatest residual scatter. Thismodel is also unable to matchthe curvature in the halos of thecold-collapse models (Fig. 10). However, it does provide very good fitsto the galaxy-sized halos, and actually has the smallest residualscatter for three of these halos (Table 3).

The shallowest, inner, negative logarithmic slope of this modeloccurs when 0 0, giving a value of 7/9 % 0:78. For nonzerovalues of0 this slope steepens roughly linearly with 0.

4.2. Sersic/Einasto Model

Sersic (1963, 1968) generalized de Vaucouleurss (1948) R1=4

luminosity profile model by replacing the exponent 1/4 with 1/n,such thatn was a free parameter that measured the shape of agalaxys luminosity profile. Using the observers notion of con-centration (see the review in Graham et al. 2001), the quantity nis monotonicallyrelated to how centrally concentrated a galaxyslight profile is. With R denoting the projected radius, Sersics R1

=n

model is often written as

I(R) Ie exp bn (R=Re)1=n 1

h in o; 18

where Ie is the (projected) intensity at the (projected) effectiveradius Re. The term bn is not a parameter but a function of n anddefined in such a way thatRe encloses half of the (projected) totalgalaxy light (Caon et al. 1993; see also Ciotti 1991, his eq. [1]). Agood approximation when nk0:5 is given in Prugniel & Simien(1997) as

bn % 2n 1=3 0:009876=n: 19

Assorted expressions related to the R1=n model can be found in

Graham & Drivers (2005) review article.Despite the success of this model in describing the luminosity

profiles of elliptical galaxies (e.g., Davies et al. 1988; Caon et al.1993; DOnofrio et al. 1994; Young & Currie 1995; Grahamet al. 1996; Graham & Guzman 2003; and references therein), itis nonetheless an empirical fitting function with no commonlyrecognizedphysical basis (but seeBinney et al.1982; Merritt et al.1989; Ciotti 1991; Marquez et al. 2001). We are therefore free toexplore the suitability of this function for describing the mass

3 The quantities and 0 are not as related as their notation suggests. Theformer is the outermost, negative logarithmic slope of the density profile, whilethe latter is the velocity anisotropy parameter at r 0.

Fig. 5.Residual profiles from application of the three-parameter (1, 3, )model (eq. [14]) to our 10 N-body density profiles. The virial radius is markedwith an arrow, and the rms residual (eq. [12a]) is inset with the residual profiles.



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density profiles, (r), of dark matter halos. Indeed, Einasto (1965,eq. [4]; 1968, eq. [1.7]; 1969, eq. [3.1]) independently developedthe functional form of Sersics equation and used it to describedensity profiles. More recent application of this profile to the mod-eling of density profiles can be found in Einasto & Haud (1989,

their eq. [14]) and Tenjes et al. (1994, their eq. [A1]). Most re-cently, the same model has been applied by Navarro et al. (2004)and Merritt et al. (2005) to characterize dark matter halos, andAceves et al. (2006) used it to describe merger remnants in simu-lated disk galaxy collisions.

Fig. 6.Residual profiles from the application of seven different parametric models (see x 4) to our cold-collapse density halos, M11 and M35. In the bottom panel,the solid curve correspondsto the two-parameter model from Dehnen & McLaughlin (2005), and the dashed curve corresponds to their three-parameter model. The rmsresidual (eq. [12a]) is inset in each figure.



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To avoid potential confusion with Sersics R1=n model, we

define the following expression as Einastos r1=n model:

(r) e exp dn (r=re)1=n 1

h in o; 20

where r is the spatial (i.e., not projected) radius. The term dn,defined below, is a function ofn such thate is the density at theradius re that defines a volume containing half of the total mass.The central density is finite and given by (r 0) ee

dn.

The integral of equation (20) over some volume gives the en-closed mass, which is also finite and equal to

M(r) 4

Zr

0

(r)r2 dr: 21

This can be solved by using the substitution x dn(r/re)1=n to

give

M(r) 4nr3e eedn d3nn (3n;x); 22

where (3n;x) is the incomplete gamma function defined by

(3n;x)

Zx0

ett3n1 dt: 23

Replacing (3n;x) with (3n) in equation (22) gives the total

mass Mtot.The value ofdn, which we first saw in equation (20), is ob-

tained by solving (3n) 2(3n; dn), where is the (complete)gamma function. The value ofdn can be well approximated (G. A.Mamon 2005, private communication) by the expression

dn % 3n 1=3 0:0079=n; for nk0:5 24

(see Fig. 11).In Paper II we recast Einastos r1

=n model using the radius r2,where the logarithmic slope of the density profile equals 2.

Einastos r1=n model (see Einasto & Haud 1989) was used in

Navarro et al. (2004, their eq. [5]) to fit their simulated dark matter

TABLE 1

Empirical Three-Parameter Models

(1, 3, ) Einasto r1=n Prugniel-Simien

Halo ID

(1)

rs(kpc)

(2)

log s(M pc

3)

(3)

(4)

(dex)

(5)

re(kpc)

(6)

log e(M pc

3)

(7)

nEin(8)

(dex)

(9)

Re(kpc)

(10)

log 0

(M pc3)

(11)

nPS(12)

(dex)

(13)

Cluster-sized Halos

A09................... 626.9 3.87 1.174 0.025 5962 6.29 6.007 (0.015) 2329 2.73 3.015 0.021

B09................... 1164 4.75 1.304 (0.037) 17380 7.66 7.394 0.041 4730 3.34 3.473 0.038C09................... 241.8 3.27 0.896 0.040 1247 4.95 3.870 0.030 738.9 2.55 2.192 (0.016)

D12................... 356.1 3.82 1.251 0.026 2663 6.02 5.939 0.020 1232 2.52 3.147 (0.019)E09................... 382.5 3.96 1.265 0.033 2611 6.06 5.801 0.032 1231 2.62 3.096 (0.030)F09................... 233.9 3.51 1.012 0.030 1235 5.26 4.280 0.025 697.3 2.63 2.400 (0.017)

Galaxy-sized Halos

G00................... 27.96 3.16 1.163 (0.020) 189.0 5.22 5.284 0.023 114.4 42.02 3.135 0.028

G01................... 35.34 3.36 1.275 0.029 252.6 5.51 5.873 (0.028) 146.0 2.01 3.425 0.032G02................... 53.82 3.59 1.229 0.034 391.4 5.74 5.725 (0.031) 214.9 2.34 3.243 0.036G03................... 54.11 3.70 1.593 0.028 405.6 5.98 7.791 (0.023) 229.1 1.47 4.551 0.024

Spherical-Collapse Halos

M11 .................. 0.0175 2.66 0.006 0.223 0.244 0.27 3.426 (0.043) 0.187 2.57 2.445 0.051

M35 .................. 0.0180 1.62 0.030 0.249 0.240 0.70 3.214 (0.059) 0.185 1.47 2.301 0.061

Notes.Col. (1): Object identification. Cols. (2)(5): (1, 3, ) model(eqs. [13] and [14]) scale radius rs, scale densitys, inner profile slope , andrms scatterof thefit.Cols.(6)(9): Einastor1

=n modelhalf-mass radius re, associated density e, profileshape nEin, andrms scatter of thefit. Cols. (10)(13):Prugniel-Simienmodelscaleradius Re, scale density

0 (the spatial density e atr Re is such thate 0eb), profile shape nPS, and rms scatter of the fit. Note that the radius and density units do

not apply to M11 and M35. For each halo, of the three models shown here, the model having the lowest residual scatter is placed in parentheses.

Fig. 7.Residual profiles from application of Burkerts two-parametermodel (eq. [15]) to our dark matter density profiles.



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halos. They obtained n % 1/(0:172 0:032) % 6 1:1. Sub-sequently, Merritt et al. (2005) showed that Einastos r1

=n modelperformed as well as the (1, 3, ) model and gave better fits forthe dwarf galaxy and galaxy-sized halos, obtaining n % 5:6 0:7. For a sample of galaxy-sized halos, Prada et al. (2006) ob-

tained similar values of 67.5.Figure 12 shows the application of equation (20) to the

N-body halos ofx 3. A comparisonwiththe (1, 3,) model fits inFigure 5 reveals that Einastos model provides a better descrip-

tion for five of the six cluster-sized halos, three of the four galaxy-sized halos, and both of the spherical-collapse halos.

Navarro et al. (2004) wrote adjusting the parameter [n] al-lows the profile to be tailored to each individual halo, resultinginimproved fits.4 Such a breaking of structural homology (see

TABLE 2

Two-Parameter Models

Burkert NFW Isotropic Dehnen-McLaughlin ( Eq. [16])

Halo ID

(1)

rs(kpc)

(2)

log 0(M pc

3)

(3)

(dex)

(4)

rs(kpc)

(5)

log s(M pc

3)

(6)

(dex)

(7)

rs(kpc)

(8)

log s(M pc

3)

(9)

(dex)

(10)

Cluster-sized Halos

A09................................. 114.0 1.65 0.242 419.8 3.50 0.042 933.7 2.43 (0.018)

B09................................. 145.2 2.23 0.247 527.2 4.03 0.068 1180.0 2.97 (0.042)C09................................. 96.16 1.74 0.181 284.4 3.42 (0.042) 554.3 2.27 0.091

D12................................. 68.39 1.62 0.230 213.3 3.34 0.051 409.1 2.17 (0.026)E09................................. 77.09 1.80 0.215 227.0 3.46 0.053 428.2 2.28 (0.037)F09................................. 80.17 1.85 0.181 229.0 3.49 (0.030) 438.2 2.32 0.066

Galaxy-sized Halos

G00................................. 10.12 1.56 0.139 22.23 2.94 (0.024) 34.43 1.59 0.037

G01................................. 10.28 1.54 0.152 23.12 2.95 0.038 36.53 1.61 (0.031)G02................................. 14.06 1.66 0.183 36.39 3.22 0.044 63.06 1.96 (0.035)G03................................. 09.35 1.32 0.179 19.54 2.68 0.066 26.98 1.23 (0.025)


M11 ................................ 0.0261 3.01 (0.203) 0.0309 2.23 0.233 0.0234 4.31 0.244

M35 ................................ 0.0265 1.98 (0.231) 0.0314 1.20 0.259 0.0236 3.29 0.269

Notes.Col. (1): Object identification. Cols. (2)(4): Burkert (1995) model scale radius rs, central density 0, and rms scatter of the fit (using m 2 in thedenominator of eq. [12a]). Cols. (5)(7): NFW (1, 3, 1) model scale radius rs, scale density 0, and rms scatter of the fit (using m 2). Cols. (8)(10): Dehnen &McLaughlin (2005, their eq. [20b]) model scale radius rs, associated density s, and rms scatter (using m 2). This model has inner and outer negative logarithmicslopes of 7/9 % 0:78 and 31/9 % 3:44, respectively. Note that the above radius and density units do not apply to M11 and M35. For each halo, the two-parametermodel with the lowest residual scatter is placed in parentheses.

Fig. 8. Residual profilesfrom application of thetwo-parameter NFW(1, 3, 1)model to our dark matter density profiles.

Fig. 9.Residual profiles from application of the two-parameter (4/9, 31/9,7/9) model (eq. [16]) from Dehnen & McLaughlin (2005, their eq. [20b]) to ourdark matter density profiles.

4 The value of n, equal to 1/ in Navarro et al.s (2004) notation, rangedfrom 4.6 to 8.2 ( Navarro et al. 2004, their Table 3).



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Graham & Colless [1997] for an analogy with projected lumi-nosity profiles) replaces the notion that a universal density pro-file may exist.

A number of useful expressions pertaining to Einastos model,when used as a density profile (eq. [20]), are given in Cardoneet al. (2005) and Mamon & xokas (2005). In particular, Cardoneet al. provide the gravitational potential, as well as approxima-tions to the surface density and space velocity dispersion of theEinasto r1

=n model, while Mamon & xokas give approximations

for the concentration parameter, central density, and Mvir/Mtot,the virial-to-total mass ratio. The nature of the inner profile slopeof Einastos r1

=n model and several other useful quantities are

presented in Paper II.

4.3. Prugniel-Simien Model: A Deprojected Sersic R1=n Model

Merritt et al. (2005) tested how well a deprojected Sersic R1=n

model fitted (r) from the Navarro et al. (2004) N-body halos.This was essentially the same as comparing the halosurface den-sities with SersicsR1

=n law. Prugniel & Simien (1997) presenteda simple, analytical approximation to the deprojected Sersic law(their eq. [ B6]):

(r) 0r

Re

pexp b

r

Re

1=n" #; 25

with

0 M

LIee

bbn(1p) (2n)

2Re n 3 p : 26

Fig. 10.Residual profiles fromapplicationof the three-parameter(3 0 )/5;(18 0)/5; 0 model (eq. [17]) from Dehnen & McLaughlin (2005, theireq. [46b]) to our dark matter density profiles.

TABLE 3

Theoretically Motivated Anisotropic Dehnen-McLaughlin

Three-Parameter Model

Halo ID

(1)

rs(kpc)

(2)

log s(M pc

3)

(3)

0

(4)

(dex)

(5)

Cluster-sized Halos

A09.................... 722.7 2.21 0.694 (0.013)B09.................... 1722 3.30 0.880 0.040

C09.................... 207.0 1.34 0.241 0.047D12.................... 322.8 1.95 0.683 0.022E09.................... 330.4 2.04 0.669 0.034

F09 .................... 193.6 1.56 0.350 0.036

Galaxy-sized Halos

G00.................... 20.89 1.11 0.422 (0.017)

G01.................... 25.88 1.28 0.568 (0.023)G02.................... 43.05 1.60 0.581 (0.027)

G03.................... 30.20 1.34 0.849 0.024

Spherical Collapse Halos

M11 ................... 0.025 4.23 0.00 0.179

M35 ................... 0.025 3.21 0.00 0.206

Notes.Col. (1): Object identification. Cols. (2)(5): Dehnen-McLaughlin(their eq. [46b]) scale radius rs, scale density s, inner profile slope

0, and rmsscatter of the fit. Note that the radius and density units do not apply to M11and M35. When the rms scatter is lower than the value obtained with the otherthree-parameter models, it is placed in parentheses.

Fig. 11.Difference between the exact value for dn from eq. (20) such that(3n) 2(3n; dn) and the two approximations inset in the figure.

Fig. 12.Residual profiles from application of Einastos r1=n model (eq. [20])

to our dark matter density profiles.



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Equation (25) is a generalization of equation (2) in Mellier &Mathez (1987), who considered only approximations to the de-projected R1

=4 law. Mellier & Mathezs model was itself a modi-fication of equation (33) from Young (1976), which derived fromthe work of Poveda et al. (1960).

In these expressions, Re, n, and b are understood to be essen-tially the same quantities that appear in the Sersic R1

=n law thatdescribes the projected density (eq. [18]). In fact, since equa-tion (25) is not exactly a deprojected Sersic profile, the correspon-dence between the parameters will not be perfect. We follow thepractice of earlier authors and define b to have the same relationto n as in equation (19). (For clarity, we have dropped the sub-scriptn from bn.) Although the parameter

0 is obtained fromfitting the density profile, it can be defined in such a way that thetotal (finite) mass from equation (25) equals that from equation(18), giving equation (26). (We stress that the n in the Prugniel-Simien profile is not equivalent to the n in eq. [20], Einastosmodel.)

This leaves the parameterp. We define p, like b, uniquely interms ofn:

p 1:0 0:6097=n 0:05463=n2: 27

Lima Neto et al. (1999) derived this expression by requiringthe projection of equation (25) to approximate as closely as pos-sible the Sersic profile with the same (Re; n), for 0:6 n 10and 102 R /Re 10

3.5 The accuracy of Prugniel & Simiens(1997) approximation, using equation (27) forp(n), is shown in

Figure 13.Terzic & Graham (2005) give simple expressions, in terms of

elementary and special functions, for the gravitational potentialand force of a galaxy obeying the Prugniel-Simien law and derivethe spatial and line-of-sight velocity dispersion profiles.

One could also allow p to be a free parameter, creating a den-sity profile that has any desired inner slope. For instance, settingp 0, the Prugniel-Simien model reduces to the Einasto model.We do not explore that idea further here.

The density at r Re is given by e 0eb, while the pro-

jected surface density atR Re, denoted byIe, can be solved forusing equation (26). Thus, one can immediately construct (a goodapproximation to) the projected mass distribution, which willhave a Sersic form (eq. [18]) with parameters (Re,Ie, and n). Thisallows the halo parameters to be directly compared with those ofSersic fits to luminous galaxies, which we do in Paper III. InPaper II we recast this model using the radius where the logarith-mic slope of the density profile equals 2.

The mass profile (Terzic & Graham 2005, their Appendix A;see also Lima Neto et al. 1999; Marquez et al. 2001) can be writ-ten as

M(r) 40R3e nbn(p3) n 3 p ;Z ; 28

where Z b(r/Re)1=n and (a;x) is the incomplete gamma func-

tion given in equation (23). The total mass is obtained by replac-ing n(3 p);Z with n(3 p) , and the circular velocity isgiven by vcirc(r) GM(r)/r 1

=2.In Figure 14 equation (25) has been applied to our dark matter

profiles. The average (plus or minus a standard deviation) of theshape parameter for the galaxy-sized and cluster-sized halos isn 3:59 0:65 and n 2:89 0:49, respectively. Merritt et al.(2005, their Table 1) found values of 3:40 0:36 and 2:99 0:49 for their sample of galaxies and clusters, respectively, in goodagreement with the results obtained here using a different set ofN-body simulations.

Figure 14 reveals that CDM halos resemble galaxies (Merrittet al. 2005), since the projection of the Prugniel-Simien modelclosely matches the Sersic R1

=n model and the latter is a good ap-proximation to the luminosity profiles of stellar spheroids. Sub-ject to vertical and horizontal scaling, CDM halos have massdistributions similar to those of elliptical galaxies with an abso-luteB-band magnitude around 18 1 mag; these galaxies haven $ 3 (see Graham & Guzman 2003, their Fig. 9). This result wasobscured until recently due to the use of different empirical mod-els by observers and modelers.

Before moving on, we again remark that we have not exploredpotential refinements to expression (27) for the quantity p, but

5 The value ofp given in eq. (27) is preferable to the value 1:0 0:6097/n 0:05563/n2 given in Marquez et al. (2000) (G. B. Lima Neto 2005, privatecommunication).

Fig. 13.Logarithmic difference between the exact deprojection of Se rsicsR1

=n model (eq. [18]) and the approximation given by Prugniel & Simien (1997)in eq. (25), using the values ofp and b given in eqs. (27) and (19), respectively.

Fig. 14.Residual profiles from application of the Prugniel-Simien model(eq. [25]) to our dark matter density profiles.



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note that this could result in a better matching of the model to thesimulated profiles at small radii. As the resolution of N-bodyclustering simulations continues to improve, it will make senseto explore such generalizations.

5. MODEL COMPARISON: WHICH DID BEST?

Table 4 summarizes how well each parametric model performedby listing the rms value of (eq. [12a]) for each set of halos,given by

rms

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

N

XNi1

2i

vuut ; 29

with N 6, 4, and 2 for the cluster-sized, galaxy-sized, andspherical-collapse halos, respectively. A detailed description ofeach models performance follows.

The bowl- and hump-shaped residual profiles associated withthe two-parameter model of Burkert (1995) reveal this modelsinability to describe the radial mass distribution in our simu-lated dark matter halos. The two-parameter model of Dehnen &McLaughlin (2005) performs considerably better, although ittoo fails to describe the cold-collapse systems and two of the sixcluster-sized halos, specifically, C09 and F09. Although this (4/9,31/9, 7/9) model never provides the best fit, it does equal oroutperform the NFW-like (1, 3, ) model in describing 3 of the12 halos (A09, D12, and G03).

In general, all of the three-parameter models perform well

(0:015 dexPP 0:04 dex) at fitting the N-body (noncollapse)halos. However, neither the (1, 3,) model nor the three-parameterDehnen-McLaughlin model can match the curvature in the den-sity profiles of the cold-collapse systems ( M11 and M35). On theother hand, both Einastos r1=n model and that from Prugniel &Simien give reasonably good fits ( $ 0:05 dex) for these twohalos.

The Prugniel-Simien model provided the best overall descrip-tion of the cluster-sized, N-body halos. The (1, 3, ) model andthe three-parameter Dehnen-McLaughlin model provided the bestfit for only one cluster-sized, N-body halo each, and even thenthe (1, 3, ) model only just outperformed the Prugniel-Simienmodel,which gave the best fit forfour of the six cluster-sized halos.For two of these halos, the size of the residual about the optimal

Prugniel-Simien fit was roughly half of the value obtained whenusing the (1, 3, ) model.

The implication of this result is that Sersics R1=n model willdescribe the projected surface density of the cluster-sized, darkmatter halos. Intriguingly, Demarco et al. (2003) and Durret et al.(2005) have observed that the (projected) hot X-ray gas distribu-tion in clusters can indeed be described with SersicsR1

=n model,although the gas can at times display a rather unrelaxed behav-ior (Statler & Diehl 2005). Studies of gravitational lensing maytherefore benefit from the use of Sersics R1

=n model for whichthe lensing equation has been solved (Cardone 2004) and forwhich numerous other properties have previously been com-puted (Graham & Driver 2005).

With regard to the galaxy-sized, N-body halos, the situation issomewhat different. Dehnen & McLaughlins (2005) anisotropicthree-parameter model provided the best fit for three of the fourprofiles, with the Einasto r1

=n model providing the best fit for thefourth profile. We also observe that Einastos model providedbetter fits than the (1, 3, ) model for three of the four N-bodyhalos. If this observation holds, namely, that the Prugniel-Simienmodel describes the density profiles of the cluster-sized halosbest, while Dehnen & McLaughlins three-parameter model pro-vides the best description of the galaxy-sized halos, it would im-ply that these halos do not have the same structural form. Ofcourse, even if the same model didprovide the best fit for bothtypes of halo, any variation in the value of the profile shape n, orcentral isotropy parameter0, would point toward the existenceof nonhomology.

While halos of different mass may be systematically better

described by different density laws, it is important to emphasizethat asingle density law provides a good fit to allof the N-bodyhalos considered here. As Table 4 shows, Einastos r1

=n law hasthe smallest, or second-smallest, value ofrms for galaxy-sized,cluster-sized, and spherical-collapse halos. None of the other pa-rametric models that we considered performs as well across theboard. The next best performer overall is the Prugniel-Simienprofile.

6. DISCUSSION

Figure 15 shows ourN-body halos, together with real ellipticalgalaxies and clusters, in the profile shape versus mass plane. Theprofile shape parameter plotted there is eithern from the SersicR1

=n model fit to the light profile, or the corresponding parameter

TABLE 4

Residual Scatter: rms Values of

Model

(1)

Cluster-sized Halos

(2)

Galaxy-sized Halos

(3)


(4)

Three-Parameter Models

Einasto............................................................ (0.028) (0.026) (0.052)

Prugniel-Simien.............................................. (0.025) 0.030 (0.056)

(1, 3, ) .......................................................... 0.032 0.028 0.236

Dehnen-McLaughlin (eq. [17])...................... 0.034 (0.023) 0.193

Two-Parameter Models

Dehnen-McLaughlin (eq. [16])...................... 0.053 0.032 0.257

NFW............................................................... 0.046 0.046 0.246

Burkert ........................................................... 0.218 0.164 0.217

Notes.Col. (1): Model. Col. (2): The rms of the six residual scatters, rms (eq. [29]), for the cluster-sized halos. Col. (3): Sim-ilar to col. (2) but for the four galaxy-sized halos. Col. (4): Similar to col. (2) but for the two spherical-collapse halos. For each halotype, the two models that perform the best are placed in parentheses.



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from the Prugniel-Simien model fit to the dark matter density.

Dynamical masses from the Demarco et al. (2003) study of gal-axy clusters are shown. We have also included the elliptical gal-axy compilation in Graham & Guzman (2003), converting theirB-band luminosities into solar masses using a stellar mass-to-light ratio of 5.3 (Worthey 1994; for a 12 Gyr old single stellarpopulation) and an absolute B-band magnitude for the Sun of5.47 B-mag (Cox 2000). This approach ignores the contributionfrom dark matter in galaxies. However, given the uncertaintieson how Mtot/L varies with L (e.g., Trujillo et al. 2004 and refer-ences therein) we prefer not to apply this correction, and notethat the galaxy masses in Figure 15 only reflect the stellar mass.

Figure 15 suggests that the simulated galaxy-sized halos havea different shape parameter, i.e., a different mass distribution, fromthe simulatedcluster-sizedhalos. The same conclusion was reached

by Merritt et al. (2005), who studied a different sample ofN-bodyhalos. The sample of dwarf galaxy and galaxy-sized halos fromthat paper had a mean (plus or minus a standard deviation) 6

profile shape n 3:04 0:34, while the cluster-sized halos hadn 2:38 0:25. We observe this same systematic difference inourN-body halos. Taking the profile shape n from the Prugniel-Simien model fits to the density profile (equivalent to the valueofn obtained by fitting Sersics R1

=n model to the projected dis-tribution), we find n 3:59 0:65 for our cluster-sized halosand n 2:89 0:49 for our galaxy-sized halos. A Studentt-test,

without assuming equal variance in the two distributions, re-veals that the above means are different at the 88% level. Ap-plying the same test to the data set of Merritt et al. (2005, theirTable 1, col. [2]), which is double the size of our sample and alsocontains dwarf galaxysized halos, we find that the means are dif-ferent at the 99.98% level. We conclude that there is a significantmass dependence in the density profiles of simulated dark matterhalos. Density profiles of more massive halos exhibit more cur-vature (smallern) on a log-log plot.

The fact thatn varies systematically with halo mass raises thequestion of which density scale and radial scale to use whencharacterizing halo structure. In the presence of a universal

density profile, the ratio between Re and r2 (the radius where thelogarithmic slope of the density profile equals 2; see Paper II)is a constant factor, but with varying values ofn this is not thecase. This remark also holds for the scale density, which is usedto measure the contrast with the background density of the uni-verse and provides the so-called halo concentration. This in turnraises the question of what concentration should actually beused, and whether systematic biases exist if one uses 2 ratherthan, for instance, e. To reiterate this point: the density ratio be-tween r r2 and r Re depends on the profile shape n, andthus, apparently, on the halo mass.

In Figure 16 we show how the use of r2 and Re produceslightly different results in the size-density diagram (e.g., Fig. 8 ofNavarro et al. 2004). The relation between size (or equivalently

mass) and central concentration (or density) varies depending onhow one chooses to measure the sizes of the halos.To better explore how the homology (i.e., universality) of CDM

halos is broken, it would be beneficial to analyze a large, low-resolution sample of halos from a cosmological cube simulationin order to obtain good statistics. Moreover, the collective impactfrom differing degrees of virialization in the outer regions, pos-sible debris wakes from larger structures, global ringing inducedby the last major merger, triaxiality, and the presence of largesubhalos could be quantified.

7. SUMMARY

We presented a nonparametric algorithm for extracting smoothand continuous representations of spherical density profiles from

6 We remind that the uncertainty on the mean is not equal to the standarddeviation.

Fig. 15.Mass vs. profile shape (1/n). For the galaxies and galaxy clusters,the shape parameters n have come from the best-fitting Sersic R1=n model to the

(projected) luminosity and X-ray profiles, respectively. The galaxy stellar massesand cluster gas masses are shown here. For dark matter halos the virial massesareshown, andthe shape parameterscome from thebest-fittingPrugniel-Simienmodel. (Note that the value of 1/n from the Prugniel-Simien model applied toa density profile is equivalent to the value of n from Sersics model applied tothe projected distribution.) We are plotting baryonic properties for the galaxiesalongside dark matter properties for the simulated halos. Filled stars, N-bodydark matterhalosfrom this paper; open plus signs, galaxy clusters from Demarcoet al. (2003); dots, dwarf elliptical (dE) galaxies from Binggeli & Jerjen (1998);triangles, dE galaxies from Stiavelli et al. (2001); open stars, dE galaxies fromGraham & Guzman (2003); asterisks, intermediate to bright elliptical galaxiesfrom Caon et al. (1993) and DOnofrio et al. (1994).

Fig. 16.Density 2, where the logarithmic slope of the density profileequals 2, plotted against (1) the radius where this occurs (open symbols) and(2) the effective radius ( filled symbols) derived from the best-fitting Prugniel-Simien model (eq. [25]). Both 2 and r2 are also computed from the best-fitting Prugniel-Simien model (see Paper II). If a universal profile existed forthese halos, the vertical difference would be constant for all halos.



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N-body data and applied it to a sample of simulated dark matterhalos. All halos exhibit a continuous variation of logarithmicdensity slope with radius; in the case of the CDM halos, thevariation of slope with radius is close to a power law. We thencompared the ability of a variety of parametric models to repro-duce the nonparametric (r) values. Over the fitted radial range0:01P r/rvir < 1, both the Einasto r

1=n model (identical in func-tional form to Sersics model but expressed in terms of space,

rather than projected, radius and density) and the Prugniel-Simienmodel (an analytical approximation to a deprojected Sersic law)provide a better description of the data than the (1, 3, ) model,i.e., the NFW-like double-power-law model with inner slope .Moreover, unlike the (1, 3, ) model, both of these models havefinite totalmass and are also capable of describing the density pro-files of halos formed from the cold collapse of a spherical over-density (Fig. 6).

The single function that provides the best overallfit to the halodensity profiles is Einastos law, equation (20):

(r) e exp dn r=re 1=n 1

h in o;

with dn defined as in equation (24). This conclusion is consistent

with that of an earlier study (Merritt et al. 2005) that was basedon a different set ofN-body halos. Typical values of the shapeparametern in equation (20) are 4P nP 7 (Table 1). Correspond-ing n-values from Sersic profile fits to the projected (surface) den-sity range from $3 to $3.5 (Fig. 15).

We propose that Einastos model, equation (20), be more widelyused to characterize the density profiles of N-body halos. As notedabove, Einastos model has already found application in a num-ber of observationally motivated studies of the distribution of massin galaxies and galaxy clusters. We propose also that the suitabilityof Einastos model for describing the luminous density profiles ofgalaxies should be evaluated, either by projecting equation (20)onto the plane of the sky or by comparing equation (20) directlywith deprojected luminosity profiles. Such a study could strengthen

the already strong connection between the density profiles of gal-axies and N-body dark matter halos (Merritt et al. 2005).

While equation (20) is a good description of all of the halomodels considered here, we found that systematic differences doexist in the best-fit models that describeN-body halos formed viahierarchical merging on the one hand, andthose formed via spher-ical collapse on the other hand, in the sense that the latter havesubstantially smaller shape parameters, n % 3:3 (Table 1). That

is, the density profiles in the cold-collapse halos decline morequickly than r3 at large radii and have shallower inner pro-file slopes than those produced in simulations of hierarchicalmerging.

With regard to just the noncollapse models, we also found sys-tematic differences between the cluster- and galaxy-sized halos.The latter are slightly better fitted by the three-parameter Dehnen-McLaughlin model, and the former are slightly better fitted by thePrugniel-Simien model (Table 4). This, together with the observa-tion that more massive halos tendto have smallershape parametersn (Fig. 15), suggests that there may not be a truly universaldensity profile that describes CDM halos.

We kindly thank Gary Mamon for his detailed comments onthis manuscript, as well as a second, anonymous referee. We arealso grateful to Walter Dehnen and Dean McLaughlin for theirhelpful corrections and comments, and to Carlo Nipoti and LucaCiotti. We also wish to thank Peeter Tenjes for tracking down andkindly faxing us copies of Einastos original papers in Russian.A.G. acknowledgessupport from NASAgrant HST-AR-09927.01-Afrom the Space Telescope Science Institute, and from the AustralianResearch Council through Discovery Project Grant DP0451426.D. M. was supportedby grants AST 04-20920 andAST 04-37519from the National Science Foundation and grant NNG 04GJ48Gfrom NASA. J. D. is grateful for financial support from the SwissNational Science Foundation. B. T. acknowledges support fromDepartment of Energy grant G1A62056.

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MERRITT ET AL.2700

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Empirical Modls for Dark Matter Halos